Question

If both the roots of the quadratic equation $x^2-2kx+k^2+k-5=0$ are less than $5$, then $k$ lies in the interval

• A. $(5,6]$
• B. $[4,5]$
• C. $k\in (-\infty ,4)$
• D. $k\in (6,\infty )$

Answer 1

The correct option is C $k\in (-\infty ,4)$

$\because$ Both the roots are less than $5$,

$\therefore \frac{2k\pm \sqrt{4k^2-4(k^2+k-5)}}{2}<5$

$\Rightarrow k\pm \sqrt{5-k}<5$

$\Rightarrow \pm \sqrt{5-k}<5-k$

$\Rightarrow \pm 1<\sqrt{5-k}$

$\Rightarrow 1<5-k$

$\Rightarrow k<4\Rightarrow k\in (-\infty ,4)$